= Solution
Regard $T_pM$ as the real <vector space> of <derivations at a point> $p$: a vector $v$ acts on germs of <smooth functions> by a linear map satisfying $v(fh)=f(p)v(h)+h(p)v(f)$. In a <coordinate chart> $x=(x^1,\ldots,x^n)$, the coordinate derivations $\partial/\partial x^i|_p$ form a <basis>, so $v=\sum_i v^i\partial/\partial x^i|_p$, with $v^i=v(x^i)$. To see that these exhaust all derivations, the first-order <Taylor theorem> locally writes $f-f(p)=\sum_i(x^i-x^i(p))h_i$ with $h_i(p)=\partial_i f(p)$. The product rule gives $v(f)=\sum_i v(x^i)\partial_i f(p)$; applying this to the coordinate functions also proves independence. Equivalently one can use velocities of smooth curves through $p$.
Set $TM=\coprod_{p\in M}T_pM$, with projection $\pi(p,v)=p$. Over a <coordinate chart> $x:U\to x(U)$ define
$$
\Phi_x:\pi^{-1}(U)\longrightarrow x(U)\times\mathbb R^n,
\qquad (p,v)\longmapsto(x(p),(v^1,\ldots,v^n)).
$$
Give the disjoint union the topology generated by the inverse images of open sets under these bijections. For overlapping charts $y=f\circ x$, the <chain rule> gives the <tangent coordinate transition>
$$
\boxed{\Phi_y\Phi_x^{-1}(z,v)=(f(z),D f(z)v).}
$$
It is a smooth bijection with smooth inverse $(w,u)\mapsto(f^{-1}(w),D f^{-1}(w)u)$. Thus the induced charts form a smooth atlas of dimension $2n$. The chain rule gives the cocycle on triple overlaps, and any additional smooth <coordinate chart> gives compatible induced charts, so the structure is independent of the starting atlas.
The total space is <Hausdorff>: different base points can be separated downstairs, and two distinct vectors over one point can be separated in a common product chart. A countable coordinate cover, together with countable bases for its products, gives <second countability>. Hence this atlas defines a <smooth manifold>. In every <local trivialization>, $\pi$ is projection onto the first factor and the fibre change $v\mapsto D f(z)v$ is an invertible linear map depending smoothly on $z$. Fibrewise addition and scalar multiplication are smooth in these product coordinates. Consequently \b[$TM\to M$ is naturally a smooth real <vector bundle> of rank $n$].
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